Page 336 - Hardware Implementation of Finite-Field Arithmetic

316 App endix A

and the corresponding architecture is
first_step: modif_csa_multiplier port map(
x => x, y => y, clk=> clk, reset => reset, start =>
start1, ps => ps, pc => pc, p => p, done => done1
);
sum <= ps + pc; mult <= sum & p;
second_step: mod_p192_reducer port map(x => mult, z => z);

The complete model also includes a control unit in charge of the
done flag generation.
The implementation results are the following (Spartan3, speed -5):

FFs LUTs Slices Period Cycles Total time
584 1,828 1,355 5.9 192 1132.8

A.5 mod p Exponentiation
The Montgomery method is used. The following values must be
previously computed:
k
16
exp_k = 2 mod p = 2 mod (2 − 2 − 1) = 2 + 1 = 16 + 1
64
64
192
192
32
16
2
exp_2k = 2 mod p = (16 + 1) = 16 + 2 · 16 + 1
2k
16
The package storing the parameter values includes the following
constant definitions:
constant K: integer := 192;
--logK is the number of bits of K
constant logK: integer := 8;
constant M: std_logic_vector(K-1 downto 0) :=
X”fffffffffffffffffffffffffffffffeffffffffffffffff”;
--minus_m = 2**k - m
constant minus_M: std_logic_vector(K downto 0) :=
‘0’ & X”000000000000000000000000000000010000000000000
001”;
constant one: std_logic_vector(K-1 downto 0) :=
conv_std_logic_vector(1, K);
--exp_k = 2**k mod m
constant exp_K: std_logic_vector(K-1 downto 0) :=
X”000000000000000000000000000000010000000000000001”;
--exp_2k = 2**(2*k) mod m
constant exp_2K: std_logic_vector(K-1 downto 0) :=
X”000000000000000100000000000000020000000000000001”;
Both the MSB-first and LSB-first algorithms have been implemented
(Spartan3, speed-5) (Table A.3).